Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves 7.1 Ellipse An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1 and r from two fixed points F1 and F, (the foci) separated by a distance of is a given positive constant. This results in the two-center bipolar coordinate equation: r1 + r = a where a is the semi-major axis and the origin of the coordinate system is at one of the foci. The corresponding parameter b is known as the semi-minor axis. The ellipse was first studied by Menaechmus, investigated by Euclid, and named by Apollonius. The focus and conic section directrix of an ellipse were considered by Pappus. In 160, Kepler believed that the orbit of Mars was oval; he later discovered that it was an ellipse with the Sun at one focus. In fact, Kepler introduced the word "focus" and published his discovery in 1609. In 1705 Halley showed that the comet now named after him moved in an elliptical orbit around the Sun (MacTutor Archive). An ellipse rotated about its minor axis gives an oblate spheroid, while an ellipse rotated about its major axis gives a prolate spheroid. Let an ellipse lie along the x-axis and find the equation of the figure given above where F1 and F are at (-c, 0) and (c, 0). In Cartesian coordinates, Bring the second term to the right side and square both sides, Now solve for the square root term and simplify 65
Square one final time to clear the remaining square root, Grouping the x terms then gives which can be written in the simple form Defining a new constant puts the equation in the particularly simple form The parameter b is called the semi-minor axis by analogy with the parameter a, which is called the semi-major axis (assuming b < a). The fact that b as defined at right is actually the semi-minor axis is easily shown by letting r1 and r be equal. Then two right triangles are produced, each with hypotenuse a, base c, and height b = a - c. Since the largest distance along the minor axis will be achieved at this point, b is indeed the semi-minor axis. If, instead of being centered at (0, 0), the center of the ellipse is at (x 0, y 0 ), at right equation becomes: 66
7. Ellipse Drawing Techniques Now we already understand circle drawing techniques. One way to draw ellipse is to use the following equation: where x 0 may be replaced by x c in case of center other than origin and same in case of y. Another way is to use polar coordinates r and θ, for that we have parametric equations: x = x c + r x cos θ y = y c + r y sin θ 7.3 Four-way symmetry Symmetric considerations can be had to further reduce computations. An ellipse in standard position is symmetric between quadrants, but unlike a circle, it is not symmetric between the two octants of a quadrant. Thus, we must calculate pixel positions along the elliptical arc throughout one quadrant, then we obtain positions in the remaining three quadrants by symmetry as shown in at right figure. 7.4 Midpoint ellipse algorithm Consider an ellipse centered at the origin: To apply the midpoint method, we define an ellipse function: f ellipse (x, y) = r y x + r x y r x r y 67
Therefore following relations can be observed: < 0, if (x, y) is inside the circle boundary f ellipse (x, y) = 0, if (x, y) is on the circle boundary > 0, if (x, y) is outside the circle boundary Now as you have some idea that ellipse is different from circle. Therefore, a similar approach that is applied in circle can be applied here using some different sampling direction as shown in the at right figure. There are two regions separated in one octant. Therefore, idea is that in region 1 sampling will be at x direction; whereas y coordinate will be related to decision parameter. In region sampling will be at y direction; whereas x coordinate will be related to decision parameter. So consider first region 1. We will start at (0, r y ), we take unit steps in the x direction until we reach the boundary between region 1 and region. Then we switch to unit steps in the y direction over the remainder of the curve in the first quadrant. At each step, we need to test the value of the slope of the curve. The ellipse slope is calculated from following equation: dy / dx = - r y x / r x y At the boundary region 1 and region, gy/ dx = -1 and r x y = r y x Therefore, we move out of region 1 whenever r y x >= r x y Figure at right shows the midpoint between the two candidate pixels at sampling position x k + 1 in the first region. Assuming position (x k, y k ) has been selected at the previous step, we determine the position along the ellipse path by evaluating the decision parameter at this midpoint: r y x + r x y r x r y =0 next P1k = f ellipse ( x k + 1, y k ½ ) f ellipse (x k +1, y k ½ ) = r y ( x k + 1) + r x ( y k ½ ) r x r y -------( 1 ) 68
If p k < 0, this midpoint is inside the ellipse and the pixel on scan line y k is closer to the ellipse boundary. Otherwise, the mid position is outside or on the ellipse boundary, and we select the pixel on scan-line y k -1. Successive decision parameters are obtained using incremental calculations. We obtain a recursive expression for the next decision parameter by evaluating the ellipse function at sampling position x k +1=x k +: f ellipse (x k +1 +1, y k+1 ½ ) = r y [( x k + 1) + 1 ] + r x ( y k+1 ½ ) r x r y ---( ) Subtracting (1) from (), and by simplification, we get P k+1 1 = P k 1 + r y ( x k + 1) + r x ( y k+1 y k ) - r x (y k+1 - y k ) + r y Where y k+1 is either y k or y k -1, depending on the sign of P k. Therefore, if P k < 0 or negative then y k+1 will be y k and the formula to calculate P k+1 will be: P1 k+1 = P k 1 + r y ( x k + 1) + r x ( y k y k ) - r x (y k - y k ) + r y P k+1 1 = P k 1 + r y ( x k + 1 ) + r y Otherwise, if P k > 0 or positive then y k+1 will be y k -1and the formula to calculate P k+1 will be: P k+1 1 = P k 1 + r y ( x k + 1) + r x ( (y k 1) y k ) - r x (y k 1 y k ) + r y P1 k+1 = P1 k + r y ( x k + 1) + r x ( y k + 1 ) - r x ( 1 ) + r y P1 k+1 = P1 k + r y ( x k + 1) r x y k + r x + r x + r y P1 k+1 = P1 k + r y ( x k + 1) r x ( y k 1 ) + r y Now a similar case that we observe in line algorithm is from where starting P k will evaluate. For this at the start pixel position will by ( 0, r y ). Therefore, putting this value is equation, we get P1 0 = r y ( 0 + 1) + r x (r y ½ ) r x r y P1 0 = r y + r x r y r x r y + ¼ r x r x r y P1 0 = r y r x r y + ¼ r x Similarly same procedure will be adapted for region and decision parameter will be calculated, here we are giving decision parameter and there derivation is left as an exercise for the students. P k+1 = P k r x ( y k + 1 ) + r x, if p k > 0 P k+1 = P k + r y ( x k + 1) r x y k + r x otherwise 69
The initial parameter for region will be calculated by following formula using the last point calculated in region 1 as: P 0 = r y (x 0 + ½ ) + r x (y 0-1 ) r x r y Since all increments are integer. Finally sum up all in the algorithm: MidpointEllipse (xcenter, ycenter, r x, r y) x =0 x =0 y = r y do DrawSymmetricPoints (xcenter, ycenter, x, y) P 0 1 = r y r x r y + ¼ r x x = x +1 If p1 k < 0 P k+1 1 = P k 1 + r y ( x k + 1 ) + r y else P k+1 1 = P k 1 + r y ( x k + 1) r x ( y k 1 ) + r y y = y -1 P 0 = r y (x 0 + ½ ) + r x (y 0-1 ) r x r y y = y -1 If p k > 0 P k+1 = P k r x ( y k + 1 ) + r x else P k+1 = P k + r y ( x k + 1) r x y k + r x x = x + 1 while ( r y x >= r x y ) 7.5 Other Curves Various curve functions are useful in object modeling, animation path specifications, data, function graphing and other graphics applications. Commonly encountered curves include conics, trigonometric and exponential functions, probability distributions, general polynomials, and spline functions. Displays of these curves can be generated with methods similar to those discussed for the circle and ellipse. We can obtain positions along curve paths directly from explicit representations y = f(x) or from parametric forms. Alternatively, we could apply the incremental midpoint method to plot curves described with implicit functions f(x,y) = 0. 70
7.6 Conic Sections A conic section is the intersection of a plane and a cone. Circle Ellipse (h) Parabola (h) Hyperbola (h) Ellipse (v) Parabola (v) Hyperbola (v) The general equation for a conic section: Ax + Bxy + Cy + Dx + Ey + F = 0 The type of section can be found from the sign of : B -4AC If B - 4AC is then the curve is a... < 0 ellipse, circle, point or no curve. = 0 parabola, parallel lines, 1 line or no curve. > 0 hyperbola or intersecting lines. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k). Equation (horiz. vertex): Equations of Asymptotes: Equation (vert. vertex): Equations of Asymptotes: Variables: Circle Ellipse Parabola Hyperbola x + y = r x / a + y / b = 1 x + y = r y / a + x / b = 1 r = circle radius a = major radius (= 1/ length major axis) 4px = y x / a - y / b = 1 y = ± (b/a)x 4py = x y / a - x / b = 1 p = distance from vertex to focus (or directrix) x = ± (b/a)y a = 1/ length major axis b = 1/ length minor axis 71
b = minor radius (= 1/ length minor axis) c = distance center to focus c = distance center to focus Eccentricity: 0 c/a c/a Relation to Focus: p = 0 a - b = c p = p a + b = c Definition: is the locus of all points which meet the condition... distance to the origin is constant sum of distances to each focus is constant distance to focus = distance to directrix difference between distances to each foci is constant 7.7 Hyperbola We begin this section with the definition of a hyperbola. A hyperbola is the set of all points (x, y) in the plane the difference of whose distances from two fixed points is some constant. The two fixed points are called the foci. Each hyperbola consists of two branches. The line segment; which connects the two foci intersects the hyperbola at two points, called the vertices. The line segment; which ends at these vertices is called the transverse axis and the midpoint of this line is called the center of the hyperbola. See figure at right for a sketch of a hyperbola with these pieces identified. Focus Center Transverse Axis Vertex Note that, as in the case of the ellipse, a hyperbola can have a vertical or horizontal orientation. We now turn our attention to the standard equation of a hyperbola. We say that the standard equation of a hyperbola centered at the origin is given by if the transverse axis is horizontal, or if the transverse axis is vertical. Notice a very important difference in the notation of the equation of a hyperbola compared to that of the ellipse. We see that a always corresponds to the positive term in the equation of the ellipse. The relationship of a and b does not determine the 7
orientation of the hyperbola. (Recall that the size of a and b was used in the section on the ellipse to determine the orientation of the ellipse.) In the case of the hyperbola, the variable in the ``positive'' term of the equation determines the orientation of the hyperbola. Hence, if the variable x is in the positive term of the equation, as it is in the equation then the hyperbola is oriented as follows: If the variable y is in the positive term of the equation, as it is in the equation then we see the following type of hyperbola: Note that the vertices are always a units from the center of the hyperbola, and the distance c of the foci from the center of the hyperbola can be determined using a, b, and the following equality: We will use this relationship often, so keep it in mind. The next question you might ask is this: ``What happens to the equation if the center of the hyperbola is not (0, 0)?'' As in the case of the ellipse, if the center of the hyperbola is (h, k), then the equation of the hyperbola becomes if the transverse axis is horizontal, or if the transverse axis is vertical. 73
A few more terms should be mentioned here before we move to some examples. First, as in the case of an ellipse, we say that the eccentricity of a hyperbola, denoted by e, is given by or we say that the eccentricity of a hyperbola is given by the ratio of the distance between the foci to the distance between the vertices. Now in the case of a hyperbola, the distance between the foci is greater than the distance between the vertices. Hence, in the case of a hyperbola, Recall that for the ellipse, Two final terms that we must mention are asymptotes and the conjugate axis. The two branches of a hyperbola are bounded by two straight lines, known as asymptotes. These asymptotes are easily drawn once one plots the vertices and the points (h, k+b) and (h, k-b) and draws the rectangle which goes through these four points. The line segment joining (h, k+b) and (h,k-b) is called the conjugate axis. The asymptotes then are simply the lines which go through the corners of the rectangle. But what are the actual equations of these asymptotes? Note that if the hyperbola is oriented horizontally, then the corners of this rectangle have the following coordinates: and Here I have paired these points in such a way that each asymptote goes through one pair of the points. Consider the first pair of points: Given two points, we can find the equation of the unique line going through the points using the point--slope form of the line. First, let us determine the slope of our line. We find this as ``change in y over change in x'' or ``rise over run''. In this case, we see that this slope is equal to 74
or simply Then, we also know that the line goes through the center (h, k). Hence, by the point-- slope form of a line, we know that the equation of this asymptote is or The other asymptote in this case has a negative slope; which is given by Using the same argument, we see that this asymptote has equation What if the hyperbola is vertically oriented? Then one of the asymptote will go through the corners of the rectangle given by Then the slope in this case will not be b/a but will be a/b. Hence, analogous to the work we just performed, we can show that the asymptotes of a vertically oriented hyperbola are determined by and 7.8 Parabola A parabola is the set of all points (x, y) that are the same distance from a fixed line (called the directrix) and a fixed point (focus) not on the directrix. See figure for the view of a parabola and its related focus 75
and directrix. Note that the graph of a parabola is similar to one branch of a hyperbola. However, you should realize that a parabola is not simply one branch of a hyperbola. Indeed, the branches of a hyperbola approach linear asymptotes, while a parabola does not do so. Several other terms exist which are associated with a parabola. The midpoint between the focus and directrix of the parabola is called the vertex and the line passing through the focus and vertex is called the axis of the parabola. (This is similar to the major axis of the ellipse and the transverse axis of the hyperbola.) See figure at right. Now let's move to the standard algebraic equations for parabolas and note the four types of parabolas that exist. As we discuss the four types, you should notice the differences in the equations that are related to each of the four parabolas. The standard form of the equation of the parabola with vertex at (0,0) with the focus lying d units from the vertex is given by if the axis is vertical and if the axis is horizontal. See figure below for an example with vertical axis and figure below for an 76
example with horizontal axis. Note here that we have assumed that It is also the case that d could be negative, which flips the orientation of the parabola. (See Figures) Thus, we see that there are four different orientations of parabolas, which depend on a) which variable is squared (x or y) and b) whether d is positive or negative. One last comment before going to some examples; If the vertex of the parabola is at (h, k), then the equation of the parabola does change slightly. The equation of a parabola with vertex at (h, k) is given by if the axis is vertical and 7.9 Rotation of Axes Note that in the sections at right dealing with the ellipse, hyperbola, and the parabola, the algebraic equations that appeared did not contain a term of the form xy. However, in our Algebraic View of the Conic Sections,'' we stated that every conic section is of the form where A, B, C, D, E, and F are constants. In essence, all of the equations that we have studied have had B=0. So the question arises: ``What role, if any, does the xy term play in conic sections? If it were present, how would that change the geometric figure?'' First of all, the answer is NOT that the conic changes from one type to another. That is to say, if we introduce an xy term, the conic does NOT change from an ellipse to a hyperbola. If we start with the standard equation of an ellipse and insert an extra term, an xy term, we still have an ellipse. So what does the xy term do? The xy term actually rotates the graph in the plane. For example, in the case of an ellipse, the major axis is no longer parallel to the x-axis or y- axis. Rather, depending on the constant in front of the xy term, we now have the major axis rotated. 77
7.10 Animated Applications Ellipses, hyperbolas, and parabolas are particularly useful in certain animation applications. These curves describe orbital and other motions for objects subjected to gravitational, electromagnetic, or nuclear forces. Planetary orbits in the solar system, for example, are ellipses; and an object projected into a uniform gravitational field travels along a parabolic trajectory. y v g Figure at right shows a parabolic path in standard position for a gravitational field acting in the negative y direction. The explicit equation for the parabolic trajectory of the object shown can be written as: x y = y 0 + a (x - x 0 ) + b ( x - x 0 ) With constants a and b determined by the initial velocity v 0 of the object and the acceleration g due to the uniform gravitational force. We can also describe such parabolic motions with parametric equations using a time parameter t, measured in seconds from the initial projection point: x = x 0 + v x0 t y = y 0 + v y0 t ½ g t Here v x0 and v y0 are the initial velocity components, and the value of g near the surface of the earth is approximately 980 cm/ sec. Object positions along the parabolic path are then calculated at selected time steps. 78
Some related real world applications are given below. 7.11 Parabolic Reflectors One of the ``real--world'' applications of parabolas involves the concept of a 3- dimensional parabolic reflector in which a parabola is revolved about its axis (the line segment joining the vertex and focus). The shape of car headlights, mirrors in reflecting telescopes, and television and radio antennae (such as the one at right) all utilize this property. In terms of a car headlight, this property is used to reflect the light rays emanating from the focus of the parabola (where the actual light bulb is located) in parallel rays. This property is used in a converse fashion when one considers parabolic antennae. Here, all incoming rays parallel to the axis of the parabola are reflected through the focus. 7.1 Elliptical Orbits At one time, it was thought that the planets in our solar system revolve around the sun in a circular orbit. It was later discovered, however, that the orbits are not circular, but were actually very round elliptical shapes. (Recall the discussion of the eccentricity of an ellipse mentioned at right.) The eccentricity of the orbit of the Earth around the sun is approximately 0.0167, a fairly small number. Pluto's orbit has the highest eccentricity of all the planets in our solar system at 0.481. Still, this is not a very large value. As a matter of fact, the sun acts as one of the foci in the ellipse. This phenomenon was first noted by Apollonius in the second century B.C. Kepler later studied this in a more rigorous fashion and developed the scientific view of planetary motion. 7.13 Whispering Galleries In rooms whose ceilings are elliptical, a sound made at one focus of the ellipse will be reflected to the other focus (across the room), allowing people standing at the two foci to hear one another very clearly. This has been called the ``whispering gallery'' effect 79
and has been used by many in the design of special rooms. In particular, St. Paul's Cathedral and one of the rooms at the United States Capitol were built with this in mind. 7.14 Polynomials and Spline Curves A polynomial function of nth degree in x is defined as y = a k x k y = a 0 x 0 + a 1 x 1 + ----------------+ a n-1 x n-1 + a n x n Where n is a nonnegative integer and the a k are constants, with a n not equal to 0. We get a quadratic when n = ; a cubic polynomial when n = 3; a quadratic when n = 4; and so forth. And obviously a straight line when n = 1. Polynomials are useful in a number of graphics applications, including the design of object shapes, the specifications of animation paths, and the graphing of data trends in a discrete set of data points. Designing object shapes or motion paths is typically done by specifying a few points to define the general curve contour, then fitting the selected points with a polynomial. One way to accomplish the curve fitting is to construct a cubic polynomial curve section between each pair of specified points. Each curve section is then described in parametric form as x = a x0 + a x1 u + a x u + a x3 u 3 y = a y0 + a y1 u + a y u + a y3 u 3 Where parameter u varies over the interval 0 to1. A curve is shown below calculated using at right equations. Continuous curves that are formed with polynomial pieces are called spline curves, or simply splines. Spline is a detailed topic; which will be discussed later in 3 dimensions. 80